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The initial space for the solution was found with techniques that had the possibility of rounding errors, and then exact proofs were found. This is a standard technique, one that I used in my PhD. So yes, it is possible to prove that each point of "contact" is exactly coincident, and if you read the paper carefully you'll see that that's what the authors did.

I was wondering this as well. Perhaps this is an example of a new, computer-assisted paradigm of math, where after extensive computation, a proposition is accepted as true if the probability of its being false is judged to be sufficiently small. I don't have the exact reference, but there was an article in some AMS publication about this in a few years ago.

I believe that there was a mathematical proof published along with the news.